Boundary $C^*$-algebras of triangle geometries

TL;DR

This paper characterizes the boundary $C^*$-algebra of a triangle geometry building of type $ ilde{A}_2$, showing it is isomorphic to a tensor product of matrix and Cuntz algebras, revealing its algebraic structure.

Contribution

It explicitly computes the boundary $C^*$-algebra for $ ilde{A}_2$ buildings with a group action, connecting geometric group actions with operator algebra structures.

Findings

The boundary $C^*$-algebra is isomorphic to a tensor product involving matrix and Cuntz algebras.

The algebraic structure reflects the geometry of the $ ilde{A}_2$ building and the group action.

Provides a concrete description of the crossed product $C^*$-algebra for these geometries.

Abstract

Let $\Delta$ be a building of type $\widetilde A_2$ and order $q$, with maximal boundary $\Omega$. Let $\Gamma$ be a group of type preserving automorphisms of $\Delta$ which acts regularly on the chambers of $\Delta$. Then the crossed product $C^*$-algebra $C(\Omega) \rtimes \Gamma$ is isomorphic to $M_{3(q+1)} \otimes\cl O_{q^2}\otimes\cl O_{q^2}$, where $\cl O_n$ denotes the Cuntz algebra generated by $n$ isometries whose range projections sum to the identity operator.